This easy do all Consider the mapping-1. L: V3 → V4 defined by L(a, b, c) = (a − b, b + c −d, a + b − 2d, b − c - d)2. L: V4 → V3 defined by L(a, b, c, d) = (2a − c, a +b+2d, a − b − c — 2d)3. L: P₂ → V4 defined by L(ax² + bx + c) = (a + b − c, a +2c, b − c, a — b)A. Prove that L is a linear transformation.B. Find the kernel of L and its dimension.C. Find the range of L and its dimension.D. Find the matrix of L with respect to the standard bases for the vector spaces involved.

According to Bartleby guidelines we can only solve 1 question or three subparts. So put other one…

1. L: V3 V4 defined by L(a, b, c) = (a - b,b+c-d, a+b-2d, b-c-d) 2. L: V4 → V3 defined by L(a, b, c, d) = (2a −c, a+b+2d, a - b - c - 2d) 3. L : P₂ → V4 defined by L(ax² + bx + c) = (a + bc, a + 2c, bc, a - b) A. Prove that I is a linear transformation. B. Find the kernel of L and its dimension. C. Find the range of L and its dimension. D. Find the matrix of L with respect to the standard bases for the vector spaces involved.

1. L: V3 V4 defined by L(a, b, c) = (a - b,b+c-d, a+b-2d, b-c-d) 2. L: V4 → V3 defined by L(a, b, c, d) = (2a −c, a+b+2d, a - b - c - 2d) 3. L : P₂ → V4 defined by L(ax² + bx + c) = (a + bc, a + 2c, bc, a - b) A. Prove that I is a linear transformation. B. Find the kernel of L and its dimension. C. Find the range of L and its dimension. D. Find the matrix of L with respect to the standard bases for the vector spaces involved.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

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