Question

X1 = [ 2 1 2], X2 = [ 1 -1 -2 ] , X3 = [ 1, 1, 1]
find the dimension and a set of basis vector of V

Answer 1

Answer on Question #63621 – Math – Linear Algebra

Question

$$X1 = [212], X2 = [1 - 1 - 2], X3 = [111]$$

find the dimension and a set of basis vector of $V$

Solution

2 1 1

1 -1 1

2 -2 1

Use the Gaussian Elimination

Divide $1^{\text{st}}$ row by 2

1 0.5 0.5

1 -1 1

2 -2 1

Subtract the $1^{\text{st}}$ row from the $2^{\text{nd}}$ row and subtract the $1^{\text{st}}$ row multiplied by 2 from the $3^{\text{rd}}$ row

1 0.5 0.5

0 -1.5 0.5

0 -3 0

Divide the $2^{\text{nd}}$ row by -1.5

1 0.5 0.5

0 1 $-\frac{1}{3}$

0 -3 0

Subtract the $2^{\text{nd}}$ row multiplied by 0.5 from the $1^{\text{st}}$ row; add the $2^{\text{nd}}$ row multiplied by 3 to the $3^{\text{rd}}$ row

1 0 $\frac{1}{3}$

0 1 $-\frac{1}{3}$

0 0 -1

Multiply the $3^{\text{rd}}$ row by (-1):

1 0 $\frac{1}{3}$

0 1 $-\frac{1}{3}$

0 0 1

Subtract the $3^{\text{rd}}$ row multiplied by $\frac{1}{3}$ from the $1^{\text{st}}$ row, add the $3^{\text{rd}}$ row multiplied by $\frac{1}{3}$ to the $2^{\text{nd}}$ row

1 0 0

0 1 0

0 0 1

The set of basis vectors is $\{E1 = [100], X2 = [010], X3 = [001]\}$ .

The dimension is $\dim = 3$ .

Answer: $\{E1 = [100], X2 = [010], X3 = [001]\}$ ; $\dim = 3$ .

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