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Q32 E

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Linear Algebra With Applications
Found in: Page 131
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Find a basis of the image of the matrix [012003000104000015000000].

The basis of the image of the matrix 012003000104000015000000 is 10000, 00100, 00010.

See the step by step solution

Step by Step Solution

Step 1:   Concept of redundant vectors; linear independence; basis

Consider vectors v1,v2,,vn n.

  1. We say that a vector vi in the list v1,v2,,vn n is redundant if vi is a linear combination of the preceding vectors v1,v2,,vi-1
  2. The vectors localid="1659362216543" v1,v2,,vn n are called linearly independent if none of them is

redundant. Otherwise, the vectors are called linearly dependent (meaning

that at least one of them is redundant).

  1. We say that the vectors v1,v2,,vn in a subspace V of n form a basis of V if they span V and are linearly independent.

Step 2:   Finding the redundant vectors of the given matrix, if any

Let us consider

Let u1=0000, u2=1000, u3=2000, u4=0100, u5=0010, u6=3450

Here we have

u1=0.u2, u3=2.u2 and u6=3u2+4u4+5u5

u1, u3, u6 are redundant vectors and u2, u4, u5 are linearly independent vectors.

Step 3: Finding basis of the image of the given matrix

Since the vectors u2, u4, u5 are linearly independent and the vectors u1, u3, u6 are redundant vectors.

Therefore, basis of the image of a matrix is equal to the linearly independent vectors.

Hence, basis of image of A is 10000, 01000, 00100.

Step 4:   Final Answer

Since the vectors u2, u4, u5 are linearly independent and the vectors u1, u3, u6 are redundant vectors

Hence, basis of image of A is 10000, 01000, 00100.

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