Given the following system of equations style=\"font-size: 12.000000pt; font-fam

Question

Given the following system of equations>

3x1 + 5x2 - 4x3 = b1>style="font-size: 12.000000pt; font-family: 'tci1'">style="font-size: 12.000000pt; font-family: 'timesnewromanpsmt'">style="font-size: 12.000000pt; font-family: 'tci2'">style="font-size: 12.000000pt; font-family: 'timesnewromanpsmt'">face="tci1" data-mce-style="font-family: tci1;">style="font-size: 12pt; ">style="font-size: 12.000000pt; font-family: 'timesnewromanpsmt'">

-3x1 - 2x2 + 4x3 = b2>style="font-size: 12.000000pt; font-family: 'tci1'">style="font-size: 12.000000pt; font-family: 'tci2'">

6x1 + x2 - 8x3 = b3>face="tci1">style="font-size: 12pt; ">style="font-size: 12.000000pt; font-family: 'timesnewromanpsmt'">style="font-size: 12.000000pt; font-family: 'tci2'">style="font-size: 12.000000pt; font-family: 'timesnewromanps'; font-style: italic">style="font-size: 12.000000pt; font-family: 'tci1'">style="font-size: 12.000000pt; font-family: 'timesnewromanpsmt'">

1.Write this as matrix equation and identify A, x and b Ax= b>style="font-size: 12.000000pt; font-family: 'tci1'">style="font-size: 12.000000pt; font-family: 'timesnewromanps'; font-style: italic">style="font-size: 12.000000pt; font-family: 'arialmt'">style="font-size: 12.000000pt; font-family: 'timesnewromanps'; font-style: italic">style="font-size: 12.000000pt; font-family: 'arialmt'">

2. Corresponding to matrix A there is a linear transformation T, what is this transformation and this transformation goes from what space to what space.>style="font-size: 12.000000pt; font-family: 'timesnewromanps'; font-style: italic">style="font-size: 12.000000pt; font-family: 'arialmt'">style="font-size: 12.000000pt; font-family: 'timesnewromanps'; font-style: italic">style="font-size: 12.000000pt; font-family: 'arialmt'">

3. What is the determinant for matrix A and what does this tell you about matrix A being invertible or not?>style="font-size: 12.000000pt; font-family: 'timesnewromanps'; font-style: italic">style="font-size: 12.000000pt; font-family: 'arialmt'">style="font-size: 12.000000pt; font-family: 'timesnewromanps'; font-style: italic">style="font-size: 12.000000pt; font-family: 'arialmt'">

4. What is a basis for the Null Space of A, what is the rank of the Null Space and what does this tell you about the linear transformation being one-to-one?>style="font-size: 12.000000pt; font-family: 'timesnewromanps'; font-style: italic">style="font-size: 12.000000pt; font-family: 'arialmt'">

5. What is the dimension of the Column Space of A and what does this tell you about the linear transformation being onto or not?>style="font-size: 12.000000pt; font-family: 'timesnewromanps'; font-style: italic">style="font-size: 12.000000pt; font-family: 'arialmt'">

Explanation / Answer

By rank-nullity theorem,

Rank of A = 3-1=2

Rank(A) = dimension of column space =2

since original transformation was R^3-->R^3, and rank is only 2 dimensional ----> the transformation

is not onto

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