another matrix representation question

[hot carbon]

Hello, I hit a road block on question 6. dont know if im overthinking it.
i wrote down an example. L(x1 x2 x3)T = {{x1 x2},{x3,x2}}. Then I got stumped finding matrix representation.

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[shrewd spoke]

The matrix representation of L doesn't multiply (x1, x2, x3)^T to make {{x1 x2},{x3,x2}}.
Rather it multiplies the coordinate representation of (x1, x2, x3) under a basis of R^3, to make a coordinate representation of {{x1 x2},{x3,x2}} under a basis of R^{2x2}.
This latter representative is an element of R^4, so we are expecting the matrix representation of L to be in R^{4x3}.

[clear vale]

Remember that ℝ^(2⨯2) is isomorphic to ℝ^4. So, you just need a nonzero transformation from ℝ^3 to ℝ^4.

[hot carbon]

thanks darpinger i think you just answered my next question since i was thinking of (4x3) x (3x1) giving a (4x1) matrix. i guess we just count that as R4
One more thing, the 2x2 matrix reuses the variables so there is still only 3 variables In it(x1,x2,and x3)
but im assuming we need 4 coefficients since the matrix representation would have 4 rows

could we use the standard basis for that like
alpha * {{1, 0}, {0, 0}} + beta * {{0, 1}, {0, 0}} + gamma * {{0, 0}, {1, 0}} + delta * {{0, 0}, {0, 1}} = {{1, 0}, {0, 0}(for x1) and rince and repeat for other variables?

[empty kernel]

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