Linear Algebra

Take the matrix with columns $1, 1$ and $-2, 0$ (Matrix $M_1$) and another matrix with columns $0, 1$ and $2, 0$ (Matrix $M_2$). Determine the matrix that represents the total effect of applying $M_1$ then $M_2$ as a single transformation. Solve this without visual aids, using only the numerical entries in each matrix.

Multiply the 2x3 matrix A with the 3x2 matrix B using the row-column rule to obtain the 2x2 matrix AB.

Given a matrix, for an element $a_{ij}$, determine its minor by excluding its row and column, and calculate its cofactor using $(-1)^{i+j} \times \text{minor}$.

Find the minor for the 3rd row and 2nd column in a given 3x3 matrix A.

Find the cofactor for the 2nd row and 3rd column in a given matrix A.

Determine the column space of the given matrix $A$.

Determine the null space of the given matrix $A$ by solving $AX = 0$.

Consider a matrix $A$ and determine the column space, which is the set of all vectors $\mathbf{b}$ such that there exists a vector $\mathbf{x}$ where $A\mathbf{x} = \mathbf{b}$.

Solve for $\mathbf{x}$ such that $A\mathbf{x} = \mathbf{0}$ to determine the null space of the matrix $A$.

Let W be the subspace of $\mathbb{R}^2$ spanned by (1, 1). Find the orthogonal projection $P_1$ from $\mathbb{R}^2$ to W and the orthogonal projection $P_2$ from $\mathbb{R}^2$ to the orthogonal complement of W.

Verify that the orthogonal projections $P_1$ and $P_2$ satisfy the properties of orthogonal projection: $P^2 = P$, $P = P^T$, $P_1 + P_2 = I$, and $P_1P_2 = P_2P_1 = 0$.

Decompose (1, 0) along (1, 1) using the orthogonal projections $P_1$ and $P_2$.

Given a vector $\mathbf{u}$ and another nonzero vector $\mathbf{a}$ in $\mathbb{R}^3$, find the vector component of $\mathbf{u}$ along $\mathbf{a}$ and the vector component of $\mathbf{u}$ orthogonal to $\mathbf{a}$.

Let B be a set of vectors $v_1, v_2, \ldots, v_k$ such that all vectors in B have length 1 $(\|\|v_i\|\| = 1)$ for all $i$ and are orthogonal to each other $(v_i \cdot v_j = 0)$ for $i \ne j$. Show that B is an orthonormal set and prove that B is also linearly independent.

Given the vectors $v_1 = \left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)$ and $v_2 = \left(\frac{2}{3}, \frac{1}{3}, -\frac{2}{3}\right)$, determine if they form an orthonormal set in $\mathbb{R}^3$.

Show that the given vectors form an orthogonal basis for $\mathbb{R}^3$. Then, express the given vector $\mathbf{w}$ as a linear combination of these basis vectors. Give the coordinates of vector $\mathbf{w}$ with respect to the orthogonal basis.

Find the dimension of the image of matrix A and the dimension of the kernel of matrix A.

Given a 2x2 matrix where one column is a linear multiple of the other, find the rank and nullity of the matrix.

Given a matrix, perform row reduction to determine the rank and nullity, ensuring the rank plus nullity equals the number of columns in the matrix.

Given a 4x5 matrix $A = \begin{pmatrix} 1 & 2 & 0 & 2 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$, find the rank and nullity of $A$.