The image contains a Linear Algebra quiz with multiple questions related to matrix operations, feature transformations in a cybersecurity context. The main task is to compute matrices E, S, T, output vector y, determinant of T, inverse of T, and verify T^-1y = x.

March 16, 2026Mathematics

ScanSolve AI Answer

March 16, 2026

Answer

\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}

Step 1: Compute $T(e_1)$ where $\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ .

T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x + 2y \\ 3x + 4y \end{pmatrix}

Substitute $x = 1$ , $y = 0$ :

T\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 + 2 \cdot 0 \\ 3 \cdot 1 + 4 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}

Step 2: Compute $T(e_2)$ where $\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ .

T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x + 2y \\ 3x + 4y \end{pmatrix}

Substitute $x = 0$ , $y = 1$ :

T\begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 + 2 \cdot 1 \\ 3 \cdot 0 + 4 \cdot 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}

Step 3: Compare $T(e_1)$ and $T(e_2)$ with the columns of $A$ .

The matrix $A$ has columns $T(e_1)$ and $T(e_2)$ :

A = \begin{pmatrix} | & | \\ T(e_1) & T(e_2) \\ | & | \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

Step 4: Verify $T(v') = A v'$ where $\mathbf{v}' = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ .

First, compute $T(v')$ :

T\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 + 2 \cdot 1 \\ 3 \cdot 1 + 4 \cdot 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}

Now compute $A v'$ :

A v' = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 + 2 \cdot 1 \\ 3 \cdot 1 + 4 \cdot 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}

Both sides match.

\begin{pmatrix 1 & 2 \ 3 & 4 \end{pmatrix}}

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Compute T(e_1) where e_1 = 1 \\ 0 . T x \\ y = x + 2y \\ 3x + 4y Substitute x = 1, y = 0: T 1 \\ 0 = 1 + 2 · 0 \\ 3 · 1 + 4 · 0 = 1 \\ 3 Step 2: Compute T(e_2) where e_2 = 0 \\ 1 .

Asked on March 16, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on March 16, 2026

Step 1: Compute T(e_1) where e_1 = 1 \\ 0 . T x \\ y = x + 2y \\ 3x + 4y Substitute x = 1, y = 0: T 1 \\ 0 = 1 + 2 · 0 \\ 3 · 1 + 4 · 0 = 1 \\ 3 Step 2: Compute T(e_2) where e_2 = 0 \\ 1 . T x \\ y = x + 2y \\ 3x + 4y Substitute x = 0, y = 1: T 0 \\ 1 = 0 + 2 · 1 \\ 3 · 0 + 4 · 1 = 2 \\ 4 Step 3: Compare T(e_1) and T(e_2) with the columns of A. The matrix A has columns T(e_1) and T(e_2): A = | & | \\ T(e_1) & T(e_2) \\ | & | = 1 & 2 \\ 3 & 4 Step 4: Verify T(v') = A v' where v' = 1 \\ 1 . First, compute T(v'): T 1 \\ 1 = 1 + 2 · 1 \\ 3 · 1 + 4 · 1 = 3 \\ 7 Now compute A v': A v' = 1 & 2 \\ 3 & 4 1 \\ 1 = 1 · 1 + 2 · 1 \\ 3 · 1 + 4 · 1 = 3 \\ 7 Both sides match. 1 & 2 \\ 3 & 4