Surjectief matrix

linear algebra - Is this matrix injective/surjective

$\begingroup$ The rank of a matrix is defined to be the dimension of the image and you correctly found that it is $2$ that it is also the dimension of $\mathbb{R}^2$ sothe image is the entire space and your matrix is surjective $\endgroup$ - Tommaso Scognamiglio Mar 23 '17 at 14:2 General Fact. Let A be a matrix and let A red be the row reduced form of A. If A red has a leading 1 in every column, then A is injective. If A red has a column without a leading 1 in it, then A is not injective. Invertible maps If a map is both injective and surjective, it is called invertible. This means, for every v in R'

  1. I saw this in a book as a Proposition but I think it's an error: Assume that the (n-by-k) matrix, A, is surjective as a mapping..
  2. Introduction to surjective and injective functions If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked
  3. ate=0, but all I got was detA=a^2 + b^2 and for surjectivity. I also know that any non square matrix cannot be bijective, so attempted to involve this in the answer, but I don't think this.

Solving for a Surjective Matrix Physics Forum

Visit our website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW Like us on Facebook: http://on.fb.me/1vWwDRc Submit your questions on Reddit. Injective, Surjective and Bijective Injective, Surjective and Bijective tells us about how a function behaves. A function is a way of matching the members of a set A to a set B A function is surjective (onto) if each possible image is mapped to by at least one argument. In other words, each element in the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection. The formal definition is the following Proof: Invertibility implies a unique solution to f(x)=y Surjective (onto) and injective (one-to-one) functions Relating invertibility to being onto and one-to-one Let's say I have a linear transformation T that's a mapping between Rn and Rm. We know that we can represent this linear transformation. Now would be a good time to return to Diagram KPI which depicted the pre-images of a non-surjective linear transformation. The vectors $\vect{x},\,\vect{y}\in V$ were elements of the codomain whose pre-images were empty, as we expect for a non-surjective linear transformation from the characterization in Theorem RPI

Surjective (onto) and injective (one-to-one) functions (video

In dit fragment wordt getoond hoe kan worden bepaald of een afbeelding injectief of surjectief is. OCW-iframe Lineaire Algebra 1 by TU Delft OpenCourseWare is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License A surjection, or onto function, is a function for which every element in the codomain has at least one corresponding input in the domain which produces that output. A function that is both injective and surjective is called bijective. Wolfram|Alpha can determine whether a given function is injective and/or surjective over a specified domain Matrix 1 is a square matrix. It's function maps 3x1 vectors into other 3x1 vectors. The matrix is not singular, meaning that all of its rows and columns are linearly independent. That implies that each value of y corresponds to 1 and only 1 value of x. That makes the function both injective and surjective. Matrix 3 is just like matrix 1 Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning

Surjection \(T\) is said to be surjective (or onto ) if its range equals the codomain. In casual terms, it means that every vector in \(W\) can be the output of \(T\) Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space A matrix transformation is onto if and only if the matrix has a pivot position in each row. Row-reduce it and then verify if the number of pivots is equal to the number of rows. Ok, with that out of the way, I need to do my rant now We prove that a linear transformation is injective (one-to-one0 if and only if the nullity is zero. The nullity is the dimension of its null space. (Linear Algebra

A function is bijective if and only if has an inverse November 30, 2015 De nition 1. Let f : A !B. We say that f is surjective if for all b 2B, there exists an a 2 minus the dimension of the range. Since T is surjective its range is R2, which has dimension two. The domain of T is also R2; thus, the dimension of the null space of T is zero. (c) The matrix representation of a linear transformation is the matrix whose columns are the images of each basis vector M[T]= 01 10 b) A is surjective (hence n ≥ k). [surjective means onto] c) dim im(A) = k. d) A∗ is injective (one-to-one). e) The columns of A span Rk. 19. Let A be a 4×4 matrix with determinant 7. Give a proof or counterexample for each of the following. a) For some vector b the equation Ax= b has exactly one solution

Proving a matrix is Bijective Physics Forum

map T : Pm(F) → Pm(F) is not surjective since polynomials of degree m are not in the range of T. 4 Homomorphisms It should be mentioned that linear maps between vector spaces are also called vector space homomorphisms. Instead of the notation L(V,W) one often sees the convention HomF(V,W) = {T : V → W | T is linear} volving the matrix of ˚. Solution note: Let Abe the matrix of T. Then T is surjective if and only if for all ~y2Rn, the system of linear equations A~x= ~yhas at least one solution (ie. is consistent). Then T is injective if and only if for all ~y2Rn, the system of linear equations A~x= ~yhas at most one solution. Then T is invertle if and only. Definition 4.6.4 If $f\colon A\to B$ and $g\colon B\to A$ are functions, we say $g$ is an inverse to $f$ (and $f$ is an inverse to $g$) if and only if $f\circ g=i_B.

Best Answer: The elementary row operations have converted T to a matrix M having reduced row echelon form. (1) Elementary row operations preserve the property of being surjective or not Injectivity and Surjectivity of the Adjoint of a Linear Map. In the following two propositions we will see the connection between a linear map $T$ being injective. Note that the dimension of the initial vector space is the number of columns in the matrix, while the dimension of the target vector space is the number of rows in the matrix. Linear transformations also exist in infinite-dimensional vector spaces, and some of them can also be written as matrices, using the slight abuse of notation known as. The effect of a shear transformation looks like 'pushing' a geometric object in a direction that is parallel to a coordinate plane (3D) or a coordinate axis (2D). In the following, the red cylinder is the result of applying a shear transformation to the yellow cylinder 29 In this shear transformation of an image of the Mon Water Quality is an especially important consideration for matrix injection. Some guidelines are available for determining the water quality parameters that you may need. Methods of analysis are available for appraising and using available matrix injection information. This field example demonstrates the philosophy of evaluating matrix injectors

[Linear Algebra] Injective and Surjective - YouTub

  1. If T: R n → R m is a one-to-one matrix transformation, what can we say about the relative sizes of n and m? The matrix associated to T has n columns and m rows. Each row and each column can only contain one pivot, so in order for A to have a pivot in every column, it must have at least as many rows as columns: n ≤ m
  2. INVERTIBLE MATRIX THEOREM MINSEON SHIN (Last edited January 31, 2014 at 7:45am.) The invertible matrix theorem (Lay, Section 2.3, Theorem 8) is a really important theorem, and it's really hard to learn and remember 12 conditions. It may be helpful to organize the conditions in the following way
  3. imum of the number of rows and the number of columns. Show that a matrix has maximal rank if and only it is either injective or surjective. Proof. (a) It suffices to show that rank(TS) ≤ rank(S) and rank(TS.
  4. IMAGE AND KERNEL OF A LINEAR TRANSFORMATION MATH 196, SECTION 57 (VIPUL NAIK) If g fis surjective, then gmust be surjective. The column vectors of the matrix.
  5. g a row operation on the n × n identity matrix, and if A is an n × m matrix, then EA is the matrix obtained by perfor
  6. 2) Find the RREF matrix for M. 3) If there is a pivot in every row of M, then T is surjective. If not, then T is not surjective. 4) If there is a pivot in every column of M, then T is injective. If not, then T is not injective. When wanting to find if a transformation is surjective and/or injective when you are given T
  7. A matrix M is then considered onto if the linear map it represents is onto. If linear map and basis don't sound familiar, here's what this means for matrices. A m×n matrix M is like a function that maps vectors from R n to R m by means of multiplication. So if v is in R n then Mv is in R m

Injective, Surjective and Bijective - Math Is Fu

Injective, surjective, and bijective transformations De nition 5 Let T: V !W be a linear transformation. It is said to be injective (or one-to-one) if for all v1, v2 2V holds v1 6=v2) T(v1) 6=T(v2); surjective (or onto) if R(T) = W; bijective if it is both injective and surjective. Slide 10 ' & $ % The null and range sets of a linear. Linear Transformations and Matrices In Section 3.1 we defined matrices by systems of linear equations, and in Section 3.6 we showed that the set of all matrices over a field F may be endowed with certain algebraic properties such as addition and multiplication Injective and Surjective Linear Maps. We will now look at two important types of linear maps - maps that are injective, and maps that are surjective, both of which terms are analogous to that of regular functions

Are there any other non-trivial *-homomorphisms between matrix algebras apart from the unitary homomorphisms? Original question: Does there exist a surjective (but not bijective) *-homomorphism between matrix algebras over the complex numbers? If so, are there any nice examples The xymatrix package The xymatrix package is included in the graphics package Xy-pic. It is possible to use Xy-pic directly for drawing commutative diagrams, 1 and some recommend its componen $\begingroup$ I think it is a legitimate question. Pietro: 1 In the matrix case, it is not generally true that $\exp(A)\exp(B)=\exp(A+B),$ so the image of the exponential map need not be a subgroup For every linear operator f in R n with standard matrix A the following conditions are equivalent: f is invertible. A is invertible. f is surjective. f is injective. Proof. Let us translate the third and the fourth conditions of this theorem into the language of matrices Surjective linear transformations Range of a linear transformation Outline 1 Surjective linear transformations Examples of surjective linear transformations Range of a linear transformation Spanning sets and surjective linear transformations Surjective linear transformations and dimension Composition of surjective linear transformations Igor V. Erovenko (UNCG) Chapter LT : Section SLT MAT 310.

Bijection, injection and surjection - Wikipedi

  1. Lecture 6 supplement: Proofs of relationships between inverses and 'jectivity. Here are a collection of proofs of lemmas about the relationships between function inverses and in-/sur-/bijectivity. This document serves at least two purposes: These proofs are good examples of what we expect when we ask you to do proofs on the homework
  2. Nandan, inverse of a matrix is related to notions of bijective, injective and surjective functions. That means you can invert a matrix only is it is square (bijective function). So a non singular.
  3. surjective), which must be one and the same by the previous factoid Proof (⇐): If it has a two-sided inverse, it is both injective (since there is a left inverse) and surjective (since there is a right inverse). Hence it is bijective
  4. matrices by H(n), the set of Hermitian positive matrices by HP(n), and the set of Hermitian positive definite matrices by HPD(n). The next lemma shows that every Hermitian positive definite matrix A is of the form eB for some unique Hermitian matrix B. As in the real case, the set of Hermitian matrices is a real vector space, but it is not a.

Determining whether a transformation is onto (video) Khan

Isomorphism and equivalence (8) De nition A function is called bijective if it is both injective and surjective. A function f : X !Y is bijective if and only if there is an invers Section SLT Surjective Linear Transformations ¶ The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, we will combine the two properties GenYoutube is a fast Youtube video downloader service. Now download videos in all formats from Youtube using GenYoutube video downloader. Using GenYoutube you can.

A First Course in Linear Algebr

Injectief of surjectief? - TU Delft OC

and that the two are not equal. The matrix for T is given by M(T) = 0 @ 0 0 0 0 1 0 0 0 0 1 A The matrix is diagonal despite the fact that T is not self-adjoint. This does not contradict the theory, since the basis used is not orthonormal. 4. Let P 2 L(V) such that P2 = P. Prove that P is an orthogonal projection if and only if P is selfadjoint Warning: Rank deficient, rank = 0... means what? . Learn more about reconstruct speech . In other words, the linear operator that the matrix represents is surjective A First Course in Linear Algebra. A First Course in Linear Algebra. Matrix Operations; Surjective Linear Transformations

WolframAlpha Examples: Injectivity & Surjectivit

Dr. Neal, WKU MATH 307 Linear Transformations from Rn to Rm Let T: Rn → Rm be a function which maps vectors from Rn to Rm.Then T is called a linear transformation if the following two properties are satisfied Z = null(A) returns a list of vectors that form the basis for the null space of a matrix A. The product A*Z is zero. size(Z, 2) is the nullity of A. If A has full. MATH 103B Homework 4 - Solutions Due May 3, 2013 (1) (Gallian Chapter 15 # 2) Prove Theorem 15.2: Let ϕ be a ring homomorphism from aring R toaringS.ThenKerϕ, defined as r R: ϕ r 0 is an ideal of R Math 225A: Di erential Topology, Homework 2 Ian Coley October 10, 2013 Problem 1.3.6. (a) If fand gare immersions, show that f gis. (b) If fand gare immersions, show that g fis. (c) If fis an immersion, show that its restriction to any submanifold of its domain is an immersion. (d) When dimX = dimY, show that immersions f : X !Y are the same as. Math 115a: Selected Solutions to HW 5 December 5, 2005 Exercise 2.4.4: Let A and B be n × n invertible matrices. Prove that AB is invertible and (AB) −1= B A . Solution: Let A and B be invertible n×n matrices

Which matrices are surjective or injective? Yahoo Answer

Surjective (onto) and injective (one-to-one) functions

De afbeelding die aan elk ooit op aarde levende mens, zijn of haar vader toevoegt (dus bijvoorbeeld (George W. Bush) = George Bush senior, (Kim Clijsters) = Lei Clijsters, enz.) is niet surjectief als afbeelding van alle mensen in alle mensen, want vrouwen treden niet op als vader. Ook als afbeelding in alle mannen is de afbeelding niet. Monday, February 14. Theorem 14.1 When a row operation is performed on an augmented matrix, the set of solutions to the corresponding matrix equation is unchanged. Therefore, the matrix equation can be solved by row-reducing the augmented matrix into a simple form where the solutions are easy to find

The matrix exponential is not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as a map from the space of all n×n matrices to the general linear group of degree n, i.e. the group of all n×n invertible matrices. Under this definition the matrix exponential is surjective Surjection. Let be a function defined on a set and taking values in a set . Then is said to be a surjection (or surjective map) if, for any , there exists an for which . A surjection is sometimes referred to as being onto. Let the function be an operator which maps points in the domain to every point in the range and let be a vector space with its standardmatrix CH T is injective ifand only if nullCA 83 2 T is surjective if and only if allA Brm Proof We will show 1 here and leave z as an exercise Assume T is injective We know that A8 8 Thus 83cnull A Let EeIR andassume that x'cnullA Then Ani 8 Thus we have Tix 167 8 Byinfectivity x 8 Thus nullIAIc 83 Assume that nullA 83 ThenAntto. some knowledge about the matrix exponential and the matrix logarithm. 5. The Exponential and the Logarithm. We rst make two observations about the matrix exponential function. Suppose we have n 1nmatrices Aand Xwhere Ais invertible. Then (AXA )m= AXmA 1. So eAXA 1 = AeXA 1 (2) by looking at the series in (1) term by term This function is not surjective. Indeed, tr(eA)=2cosω when a2 +bc < 0, tr(eA)=2coshω when a2 +bc > 0, and tr(eA) = 2 when a2 +bc =0.As a consequence, for any matrix A with null trace, tr eA ≥−2, and any matrix B with determinant 1 and whose trace is less than −2is not the exponential eA of any matrix A with null trace. For example, B.

The Matrix of a Linear Transformation Finding the Matrix A function if surjective (onto) if every element of the codomain has a preimage in the domain. That is, For everyb in B there is some a in A such that f(a) = b. That is, the codomain is equal to the range/image Jonathan Ch´ave Proof. To a), note that ϕ is surjective since inverses exist in a group, and ϕ is injective since inverses are unique. To b), note that since ϕ : G → G is a bijection, to prove it's an isomorphism it suffices to show it's a homomorphism. To that end, note that if ϕ is a homomorphism then for a,b ∈ G we hav A function f from A to B is called onto, or surjective, if and only if for every element b ∈ B there is an element a ∈ A with f(a) = b. A function is called a surjection if it is onto. A function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto

Surjection, Injection, Bijection - people

Consider B' a g(q, s) x q binary s-surjective matrix, and then proceed as before. Then c3 decreases to only 10256. Applications of coding theory 245 Having found good s-surjective families, let us turn back to defective memories, that is to condition P2. Namely how can we find small s-surjective families with a small information set to ±1 ∈ Z3) value of the determinant, we obtain a matrix with the determinant = 1. Thus the order of the group of such matrices is 48/2 = 24. Compute the determinant of the n×n-matrix with all entries on the diagonal equal to 2, right under and right above the diagonal −1, and 0 everywhere else. Let ∆n denote the determinant. Using the. MATH 311: COMPLEX ANALYSIS | AUTOMORPHISM GROUPS LECTURE 3 Proof. Since the Riemann sphere is compact, fcan have only nitely many poles, for otherwise a sequence of poles would cluster somewhere, giving a nonisolate Math 210C. A non-closed commutator subgroup 1. Introduction In Exercise 3(i) of HW7 we saw that every element of SU(2) is a commutator (i.e., has the form xyx 1y 1 for x;y2SU(2)), so the same holds for its quotient SO(3). We have seen in Exercise 2 of HW8 that if Gis a connected compact Lie group then its commutator subgroup G0is closed We have seen that certain common relations such as =, and congruence (which we will deal with in the next section) obey some of these rules above. The relations we will deal with are very important in discrete mathematics, and are known as equivalence relations. They essentially assert some kind of equality notion, or equivalence, hence the nam

Honors Algebra 4, MATH 371 Winter 2010 Solutions 1 1. Let R be a ring. An element x of R is called nilpotent if there exists an integer m ≥ 0 such that xm = 0. (a) Show that every nilpotent element of R is either zero or a zero-divisor. (b) Suppose that R is commutative and let x,y ∈ R be nilpotent and r ∈ R arbitrary. Prov Linear algebra explained in four pages Excerpt from the NO BULLSHIT GUIDE TO LINEAR ALGEBRA by Ivan Savov Abstract—This document will review the fundamental ideas of linear algebra. We will learn about matrices, matrix operations, linear transformations and discuss both the theoretical and computational aspects of linear algebra. Th Controllability and Observability In this chapter, we study the controllability and observability concepts. Controllability is concerned with whether one can design control input to steer the state to arbitrarily values. Observability is concerned with whether without knowing the initial state, one can determine the state of a syste Using the Jordan canonical form, show that every complex square matrix is the limit of a sequence of diagonalizable matrices

matrix M by a matrix E which is formed by multiplying the ithrow of the n nidentity matrix by . (iii) The operation of adding times the jthrow of M to the ithrow of M can be reproduced by multiplying the matrix M by a matrix E + j formed by adding times the jthrow of the n nidentity matrix to the ithrow of that identity matrix. 1.1.2 Note that a square matrix A is injective (or surjective) iff it is both injective and surjective, i.e., iff it is bijective. Bijective matrices are also called invertible matrices, because they are characterized by the existence of a unique square matrix B (the inverse of A, denoted A−1) such that AB = BA= I. 2 Trace and determinan Surjective function's wiki: In mathematics, a function f from a set X to a set Y is surjective (or onto ), or a surjection , if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f ( x ) = y

Bilinear forms and their matrices Joel Kamnitzer March 11, 2011 0.1 Definitions A bilinear form on a vector space V over a field F is a map H : V ×V → The matrix A=[1,2;2,1;1,1] (three rows and two columns) induces a linear map from R 2 to R 3, with domain R 2. Synonyms: If a linear transformation T is represented by a matrix A, then the range of T is equal to the column space of A. Rank The rank of a matrix is the dimension of the row space, which is equal to the dimension of the column space

How to know if a matrix is onto - Quor

Start studying Linear Algebra II Final Definitions and Theorems. Learn vocabulary, terms, and more with flashcards, games, and other study tools 2] The matrix equation y = Ax where A is an mxn matrix and x and y are vectors from two different vector spaces defines a function from one vector space into another. The domain consists of vector space V and the co-domain consists of vector space W with x in V and y in W. Matrix A represents the function which can be viewed as an operator.

The Kronecker product will then be employed to solve linear matrix equations. An investigation of the commutativity of the Kronecker product will be carried out using permutation matrices. The Jordan -Canonical form of a Kronecker product will be examined. Variations such as the Kronecker sum and generalized Kronecker product will be introduced a matrix ABis invertible if and only if the linear transformation L ABassociated with the matrix ABis invertible, which means that R(AB) = Vand N(AB) = f0g (surjective and injective). Let v2V. Since Bis invertible, there exists u2Vsuch that v= B(u). Since Ais invertible, there exists w2Vsuch that u= A(w). Hence v= AB(u). Therefore, ABis surjective A Rotation matrix is given by: R(theta) = [ ] Show that (R(theta))-1 = R(-theta) for any angle of rotation (theta) A Unitary Matrix is a matrix, which satisfies U-1 = UT. Show, using the result of (a), that the Rotation Matrix is a Unitary Matrix If we perform two rotations R(theta1) and R(theta2) show that the order in which we perform the.

Surjective function In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y Matrix Representations of Linear Transformations and Changes of Coordinates 0.1 Subspaces and Bases 0.1.1 De nitions A subspace V of Rnis a subset of Rnthat contains the zero element and is closed under additio Free step-by-step solutions to Linear Algebra & Differential Equations (Custom Edition for University of California, Berkeley) (9781256873211) - Slade Systems of Linear Equations Vectors Matrices Vector Spaces Determinants Eigenvalues Linear Transformations Representations Eigenvalues, Eigenvectors, Representations Row Space of a Matrix is a Subspace Determinant of Matrices of Size Two Composition of Surjective Linear Transformations is Surjective Linear Transformation Nonsingular Matrix Equivalences, Round 2 Column Vector Solution of a.

Gis not the trivial group and it is not surjective if His not the trivial group. 4. The determinant det: GL n(R) !R is a homomorphism. This is the content of the identity det(AB) = detAdetB. Here det is surjective, since , for every nonzero real number t, we can nd an invertible n n matrix Amuch that detA= t. For example, one can take Ato b Codomain vs Range. The Codomain and Range are both on the output side, but are subtly different. The Codomain is the set of values that could possibly come out. The Codomain is actually part of the definition of the function. And The Range is the set of values that actually do come out H to be surjective, in terms of the matrix H? • L H is surjective iff H is full-rank. Justification: • Suppose L H is surjective. [We verify that H is full-rank.] Pick any b∈ Rm. [We verify that bis a real-linear combination of the columns of H.] Since L H is surjective, there exists some x∈ Rn such that b= L H(x). Note that b= Hx Under this definition the matrix exponential is surjective for complex matrices, although still not surjective for real matrices. The projection from a cartesian product A × B to one of its factors is surjective. In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function Download Citation on ResearchGate | Surjective maps on matrices preserving the local spectral radius distance | Let [Inline formula] be a non-zero vector in [Inline formula]. We prove that if.

Recommended Citation. KHACHORNCHAROENKUL, PRATHOMJIT and Pianskool, Sajee. (2019), Surjective Additive Rank-1 Preservers on Hessenberg Matrices, Electronic Journal of Linear Algebra, Volume 35, pp. 24-34 I. MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in , and for some , then either or is not invertible. Example of a Group that is Not a Matrix Lie Group Let where . Then there exists ! such that # $ , but is invertible. Thus is not a matrix Lie group. Examples of Matrix Lie.

Its inverse, the exponential function, is not surjective as its range is the set of positive real numbers and its domain is usually defined to be the set of all real numbers. The matrix exponential is not surjective when seen as a map from the space of all n×n matrices to itself matrix. TRUE The columns on the identity matrix are the basis vectors in Rn. Since every vector can be written as a linear combination of these, and T is a linear transformation, if we know where these columns go, we know everything. I If T : R2!R2 rotates vectors about the origin through an angle ˚, then T is a linear transformation. TRUE. To. Non-Elementary Surjective Dimension Representations Nicholas Long June 9, 2010 Abstract The dimension representation has been a useful tool in studying the mys-terious automorphism group of a shift of finite type, the classification of shifts of finite type, and surrounding problems. However, there has been lit Math 304 Answers to Selected Problems 1 Section 4.2 5. Findthestandardmatrixrepresentationsforeachofthefollowinglinear operators. (a) L is the linear operator that. 2.Matrix of a given transformation. 3.Subspaces associated to a Linear Transformation. 4.Injective (one-to-one) and Surjective (onto) transformations. 5.Transformation with desired properties. 2 A Linear Transformation. 1.A Linear Transformation extends the idea of a function so that the domain is <n rather than just the eld of real numbers