Linear Algebra

Solve the given system of two equations with two variables using the Gaussian elimination method.

Given a transformation that projects the space onto the z-axis, where did the vectors in $\mathbb{R}^3$ go, and are there any vectors that were mapped to the zero vector?

For the transformation, identify all vectors that fall on the line $y = -2x$ and are transformed to the zero vector, and describe the image set which consists of vectors that land on the line $y = x$.

Explain the concept of Hilbert space in the context of quantum mechanics.

Consider the linear transformation that maps $\mathbb{R}^3$ to $\mathbb{R}^2$ given by $L(v) = (v_1, v_2 - v_3)$. Find the image of the subspace of $\mathbb{R}^3$ given by the vectors of length 3 where the second element is two times the first element, and the third element is 0.

Consider the linear transformation that maps $\mathbb{R}^3$ to $\mathbb{R}^2$ given by $L(v) = (v_1, v_2 - v_3)$. Find the kernel of this transformation.

Find the kernel and range of the differentiation operator on $P_2$. Specifically, for the linear transformation $D$ from the vector space of polynomials of degree two or less ($P_2$) to the vector space of polynomials of degree one or less ($P_1$), determine the kernel and range.

Find the quadratic equation through the origin that is a best fit for these three points: $(1, 1)$, $(2, 5)$, and $(-1, -2)$.

Find the vector $\hat{x}$ such that $A\hat{x}$ is the closest to $b$ using the least squares approximation.

Find the least squares approximating line for the set of four points (1, 3), (2, 4), (5, 5), and (6, 10).

Imagine that you have a set of data points in $x$ and $y$, and you want to find the line that best fits the data. This is also called regression.

Using the least squares method, solve for the best-fit line given a set of data points.

Given vectors $V_1 = \begin{bmatrix} 1 \\ -2 \\ -5 \end{bmatrix}$, $V_2 = \begin{bmatrix} 2 \\ 5 \\ 6 \end{bmatrix}$, and $B = \begin{bmatrix} 7 \\ 4 \\ -3 \end{bmatrix}$, determine if there exist scalars $x_1$ and $x_2$ such that $x_1 V_1 + x_2 V_2 = B$.

Are the vectors $\mathbf{v}_1 = (1, 2, 3)$, $\mathbf{v}_2 = (2, -1, 4)$, and $\mathbf{v}_3 = (0, 5, 2)$ linearly independent?

Find an $X$ in $\mathbb{R}^2$ so that the image of $X$ is $\begin{pmatrix} 3 \\ 2 \\ -5 \end{pmatrix}$.

Is the vector $\begin{pmatrix} 3 \\ 2 \\ 5 \end{pmatrix}$ in the range of $T$?

Determine the conditions on $a$ and $b$ that make a matrix $A$ invertible. Additionally, find $A^{-1}$ when it is invertible.

Given matrix A which is [3, 1, 4] and matrix B which is [4, 2; 6, 3; 5, 8], multiply matrix A by matrix B, and determine the size and elements of the resultant matrix.

Given matrix A which is $\begin{bmatrix}3 & 4 \\ 7 & 2 \\ 5 & 9\end{bmatrix}$ and matrix B which is $\begin{bmatrix}3 & 1 & 5 \\ 6 & 9 & 7\end{bmatrix}$ , can matrix A be multiplied by matrix B? If so, find the size and elements of the resultant matrix.

Describe the effects of applying one transformation and then another, specifically using the example of first rotating the plane 90 degrees counterclockwise, then applying a shear. Determine the overall effect as a single linear transformation and express it with a matrix.