Question

QUESTION 3. Consider the following set of vectors: T= {(4,4,0,8), (-8,0,40, 24), (14,0,0,7), (26,4, -40, -9)} Find the dimension of the subspace spanned by T. Carefully explain your method. (6 marks)

Answer 1

Given (-8, 0,40, 24), T= {(4 4 08 (14,0,0,7), (26,4, -40,975 We have to find dimension I nension of subspace spanned banned by T. front of all we know that the standard basis of 1R4 will be of the form, {(1,0,0,0), (0,1,0,0), (0.0, 1.0)(0,0,0,1)9 --* Now, using T, we will create a matrix and by how operations we will create a row- echolon form, where the elements will be like -- (*). Then the non zero rows of now echolor matrix will show us the basis of the subspace, by the vectons of T. After getting basis, we will able to find dime subspace shar spanned by T. Given matrox form: ( 4 4 0 8 1-8 0 40 24 14 0 0 7 126 4 40 -9 / R 3 R 2 1 02 PU dimension 40 24 - - 0 - ut o -00 I R37 R 2 N 0 0 26 4 - 40 - 9 R R2 + R R Rg - 24 Ra > R4-26 Ri 1 1 10 1 0 -2 lo -22 0 2 5 5 0 -3 -40 -61

RR - R₂ 2 2 - + 28, Ra Ra +2222 т о -5 -3 от 55 oo 107 oo 10 49/ р, 4 В. - С. 23 o -5 - от 5 5 0 0 10 - oo 10 - г о -5 - от 5 5 oo 107 ооо 1 o o , - от л two otwtwo R/+R + I R3 г., , ; 3) o o o 31- и одо о 10 + оооо р, , , - - 3 , Е, ВТ, R. , o o . To o o 31, о о т 1/to o o oo, o o o o o o (. оого o o o o - Е, 3 - 1 o o o 1 = A o o o | о о го o o o o

In matmx A, we find three non zero rows where the vectors are R (1,0,0,0), (0, 0, 0,1),(0,0,1,0) look So, they are linearly independent. so, basis of T = {(1,4,08) (-8,0,40, 24), (14,0,0,7) we can see, (26, 4, 40,-9) can be written - with the help of basis of t. 10(4,4,9, 8) + (1)•(-8,0,40, 24) + I. (14,0,0,1) - 4 + 8 +14, 4-0 +0, 0-40+0, 8-24 +7) = ( 26, 4, -40, -9) (Proved) so, our basis is correct since the basis consists of 3 vectons, dimension will be? 3 (Rus)
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