Consider T = is a spanning set for the image of T. Suppose that for every linearly independent set T in V, L( T) is linearly independent in W. Let L: V → W be a linear transformation between vector spaces. Use this fact to prove that if x 1, …, x n + 1 ∈ R, with x 1,…, x n+1 distinct, then for any given a 1, …, a n + 1 ∈ R, there is a unique polynomial p ∈ P n such that p( x 1) = a 1, p( x 2) = a 2,…, p( x n) = a n, and p( x n+1) = a n+1. Recall from algebra that a nonzero polynomial of degree n can have at most n roots. (b)įor each choice of x 1, x 2, x 3, a, b, c ∈ R, show that the polynomial p from part (a) is unique. Use an argument similar to that in Example 4 to show that for any given a, b, c ∈ R, there is a polynomial p ∈ P 2 such that p( x 1) = a, p( x 2) = b, and p( x 3) = c. Let x 1, x 2, x 3 be distinct real numbers. This exercise, related to Example 4, concerns roots of polynomials.
(b)Įxplain why L does not contradict Corollary 5.13. Suppose that L: M 22 → P 3 is a linear transformation and that L is not one-to-one.
Suppose that L: R 6 → P 5 is a linear transformation and that L is not onto. This exercise explores the concepts of one-to-one and onto in certain cases. Prove that L is one-to-one but is not onto. Let A be a fixed n × n matrix, and consider L: M n n → M n n given by L( B) = AB − BA. Show there is no one-to-one linear transformation from R m to R n. Show there is no onto linear transformation from R n to R m. Which of these linear transformations are one-to-one? Which are onto? Justify your answers by using row reduction to determine the dimensions of the kernel and range. In each of the following cases, the matrix for a linear transformation with respect to some ordered bases for the domain and codomain is given. L: R 2 → R 2 given by L x 1 x 2 = − 4 − 3 2 2 x 1 x 2 (b) Which of the following linear transformations are one-to-one? Which are onto? Justify your answers by using row reduction to determine the dimensions of the kernel and range. L: P 2 → M 22 given by L ( a x 2 + b x + c ) = a + c 0 b − c − 3 a 2. L: M 23 → M 22 given by L a b c d e f = a − c 2 e d + f ★(h) L: M 22 → M 22 given by L a b c d = d b + c b − c a ★(g) L: P 2 → P 2 given by L( ax 2 + bx + c) = ( a + b) x 2 + ( b + c) x + ( a + c) (f) L: P 3 → P 2 given by L( ax 3 + bx 2 + cx + d) = ax 2 + bx + c ★(e) (b) Find the rank and nullity of the matrix $A$ in part (a).Which of the following linear transformations are one-to-one? Which are onto? Justify your answers without using row reduction. Find a Basis of the Range, Rank, and Nullity of a Matrixįind a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.
The matrix representation of the linear transformation $T$ is given by Linear Algebra Midterm Exam 2 Problems and Solutions.