Is the function T[x1 x2 x3]=det [2 x1 5 3 x2 6 4 x3 7] from R3 to R a linear transformation? Here we are placing the input variables x1, x2, x3 in the second column, choosing arbitrary constants for all the other entries.

Question

Is the function

[latex]T\begin{bmatrix}x_1\x_2\x_3\end{bmatrix}=det\begin{bmatrix}2&x_1&5\3&x_2&6\4&x_3&7\end{bmatrix}[/latex]

from [latex]ℝ^3[/latex] to ℝ a linear transformation? Here we are placing the input variables [latex]x_1, x_2, x_3[/latex] in the second column, choosing arbitrary constants for all the other entries.

Accepted Answer

Note that

[latex]T\begin{bmatrix}x_1\x_2\x_3\end{bmatrix}=det\begin{bmatrix}2&x_1&5\3&x_2&6\4&x_3&7\end{bmatrix}= (6 · 4 − 3 · 7)x_1 + (2 · 7 − 5 · 4)x_2 + (5 · 3 − 2 · 6)x_3[/latex]

[latex]= 3x_1 − 6x_2 + 3x_3[/latex].

Therefore, T is a linear transformation, by Definition 2.1.1, since the output is a linear combination of the input variables.

Question 6.1.6: Is the function T[x1 x2 x3]=det [2 x1 5 3 x2 6 4 x3 7] from ......

Related Answered Questions

Use Laplace expansion to compute det A for A=[1 0 1 2 9 1 3 0 9 2 2 0 5 0 0 3]. ...

Verified Answer:

Find det[0 7 5 3 1 1 2 1 1 1 2 −1 1 1 1 2]. ...

Verified Answer:

Consider the matrices A[1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9] and B=[6 7 8 9 8 1 2 3 4 5 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9]. Note that B is obtained from A by swapping the first two rows. Express det B in terms of det A. ...

Verified Answer:

Let A=[1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9]. Express det(A^T ) in terms of det A. You need not compute det A. ...

Verified Answer:

Find det M for M=[a11 a12 b11 b12 a21 a22 b21 b22 0 0 c11 c12 0 0 c21 c22]. ...

Verified Answer:

Find det A for A=[6 0 1 0 0 9 3 2 3 7 8 0 3 2 9 0 0 4 0 0 5 0 5 0 1]. ...

Verified Answer:

Find det A for A=[0 2 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 0 0 0 5 0 0 0 1 0 0 0]. ...

Verified Answer:

Apply Definition 6.1.3 to a 2 × 2 matrix, and verify that the result agrees with the formula given in Theorem 2.4.9a. ...

Verified Answer:

For three column vectors u , v , w in R3 , what is the relationship between the determinants of A = [u v w] and B =[ u w v ]?Note that matrix B is obtained by swapping the last two columns of A. ...

Verified Answer:

Find det A for A = [1 2 3 4 5 0 2 3 4 5 0 0 3 4 5 0 0 0 4 5 0 0 0 0 5]. ...

Verified Answer: