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Determine if b is a linear combination of the other vectors. if so, express b as a linear combination. (if b cannot be written as a linear combination of the other two vectors, enter dne in both answer blanks.) a1 = −2 5 , a2 = 7 −3 , b = 32 7

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icedipping
b = 5 * a1 + 6 * a2

You can either solve this by solving the simultaneous equations directly or by matrices.

Using a system of equations, it is found that there is a linear combination, and it is:

[tex]b = -3.54a_1 + 3.56a_2[/tex]

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Vector b is a linear combination of vectors [tex]a_1[/tex] and [tex]a_2[/tex] if there are values of x and y such that:

[tex]xa_1 + ya_2 = b[/tex]

Then:

[tex]x(-2,5) + y(7,-3) = (32,7)[/tex]

The system is:

[tex]-2x + 7y = 32[/tex]

[tex]5x - 3y = 7[/tex]

Multiplying the first equation by 5, second by -2:

[tex]-10x + 35y = 160[/tex]

[tex]-10x + 6y = -14[/tex]

Adding them:

[tex]41y = 146[/tex]

[tex]y = \frac{146}{41}[/tex]

[tex]y = 3.56[/tex]

Then, for x:

[tex]-2x + 7y = 32[/tex]

[tex]2x = 7y - 32[/tex]

[tex]x = \frac{7y - 32}{2} = \frac{7(3.56) - 32}{2} = -3.54[/tex]

Thus, there is a linear combination, and it is:

[tex]b = -3.54a_1 + 3.56a_2[/tex]

A similar problem is given at https://brainly.com/question/21278636