Question

Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear map defined by \[ T(x_1, x_2, x_3) = (3x_1 + 5x_2 + x_3, x_3, 2x_1 + 2x_3). \] {Then the rank of \( T \) is equal to _______ (answer in integer).}

Hint:

To find the rank of a matrix, reduce it to row echelon form and count the number of non-zero rows. The rank is equal to the number of linearly independent rows.

Answer 1

Correct Answer: 3

To find the rank of the linear map \( T \), we determine the number of linearly independent rows in its matrix representation. The map \( T \) is given by: \[ T(x_1, x_2, x_3) = \begin{pmatrix} 3 & 5 & 1 \\ 0 & 0 & 1 \\ 2 & 0 & 2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}. \] Thus, the matrix representation of \( T \) is: \[ A = \begin{pmatrix} 3 & 5 & 1 \\ 0 & 0 & 1 \\ 2 & 0 & 2 \end{pmatrix}. \] Now, we find the rank of this matrix by reducing it to row echelon form (REF). Subtract \( \frac{2}{3} \) of the first row from the third row: \[ \begin{pmatrix} 3 & 5 & 1 \\ 0 & 0 & 1 \\ 0 & -\frac{10}{3} & \frac{4}{3} \end{pmatrix}. \] Multiply the third row by \( -\frac{3}{10} \): \[ \begin{pmatrix} 3 & 5 & 1 \\ 0 & 0 & 1 \\ 0 & 1 & -\frac{2}{5} \end{pmatrix}. \] Now, all three rows are non-zero and linearly independent. Thus, the matrix has 3 linearly independent rows. Rank of \( T \) is: \[ \boxed{3} \]