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Math
Posted about 1 year ago

In the problems below, S={1,2,3,4}S=\{1,2,3,4\}.
Calculate the matrix M(T)\mathrm{M}(T) representing the given linear transformations (relative to standard bases)
T:R4Fun(S)T: \mathbb{R}^{4} \rightarrow \operatorname{Fun}(S) defined by
T(a,b,c,d)=a(3χ1+χ22χ3)+b(χ12χ3)+c(χ12χ3χ4)
T(a, b, c, d)=a\left(3 \chi_{1}+\chi_{2}-2 \chi_{3}\right)+b\left(\chi_{1}-2 \chi_{3}\right)+c\left(\chi_{1}-2 \chi_{3}-\chi_{4}\right)
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Answer from Sia
Posted about 1 year ago
Solution by Steps
step 1
To find the matrix M(T)M(T) representing the linear transformation TT, we need to apply TT to the standard basis vectors of R4\mathbb{R}^4
step 2
The standard basis vectors in R4\mathbb{R}^4 are e1=(1,0,0,0)e_1 = (1, 0, 0, 0), e2=(0,1,0,0)e_2 = (0, 1, 0, 0), e3=(0,0,1,0)e_3 = (0, 0, 1, 0), and e4=(0,0,0,1)e_4 = (0, 0, 0, 1)
step 3
Apply TT to e1e_1: T(1,0,0,0)=3χ1+χ22χ3T(1, 0, 0, 0) = 3\chi_1 + \chi_2 - 2\chi_3. This gives us the first column of M(T)M(T): (3,1,1,0)(3, 1, 1, 0)
step 4
Apply TT to e2e_2: T(0,1,0,0)=χ12χ3T(0, 1, 0, 0) = \chi_1 - 2\chi_3. This gives us the second column of M(T)M(T): (0,0,0,0)(0, 0, 0, 0)
step 5
Apply TT to e3e_3: T(0,0,1,0)=χ12χ3χ4T(0, 0, 1, 0) = \chi_1 - 2\chi_3 - \chi_4. This gives us the third column of M(T)M(T): (1,2,2,0)(1, -2, -2, 0)
step 6
Apply TT to e4e_4: T(0,0,0,1)=χ4T(0, 0, 0, 1) = -\chi_4. This gives us the fourth column of M(T)M(T): (2,0,1,0)(-2, 0, -1, 0)
step 7
Combine the results from steps 3 to 6 to form the matrix M(T)M(T):
$$ M(T) = \begin{pmatrix} 3 & 0 & 1 & -2 \\ 1 & 0 & -2 & 0 \\ 1 & 0 & -2 & -1 \\ 0 & 0 & 0 & 0 \end{pmatrix} $$
Answer
The matrix M(T)M(T) representing the linear transformation TT is: M(T)=(3012102010210000) M(T) = \begin{pmatrix} 3 & 0 & 1 & -2 \\ 1 & 0 & -2 & 0 \\ 1 & 0 & -2 & -1 \\ 0 & 0 & 0 & 0 \end{pmatrix}
Key Concept
Matrix representation of a linear transformation
Explanation
The matrix representing a linear transformation with respect to the standard bases is constructed by applying the transformation to each standard basis vector and using the resulting vectors as the columns of the matrix.

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