$two short Linear Algebra questions$

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I need to have a solution that fits the correct one, and explanation of why there correct answer is the correct one. I attached the image with both questions.

Linear Algebra

Ugher

Matchmaticians two short Linear Algebra questions  File #1

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Answer 1

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For convenience, I'll write the vectors as rows.

$(5,2) = 5e_1 + 2e_2$, so $T(5,2)=5T(e_1)+2T(e_2)=5*(-1,2)+2*(3,2)=(-5+6,10+4)=(1,14)$.

Similarly, $(1,-1)=e_1-e_2$, so $T(1,-1)=T(e_1)-T(e_2)=(-1,2)-(3,2)=(-1-3,2-2)=(-4,0)$.

For the second question, matrix multiplication is row by column, so you can multiply a $5 \times 3$ matrix by columns of length $5$ (so the dimension of the domain is $5$), and get columns of length $3$ (so the dimension of the codomain is $3$).

More easily, the image of a matrix is the span of the columns, and so the dimension of the image is the height of the matrix.

Alessandro Iraci

Ugher

thank you for your help, I may post questions of the same nature next week
Alessandro Iraci

You're welcome!

$two short Linear Algebra questions$

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