### Find an explicit description of nul A by listing vectors that span the null space.

begin{equation*} A = begin{bmatrix} 1 & 2 & 3 & -7 \ 0 & 1 & 4 & -6 end{bmatrix} end{equation*} This problem aims to find the vectors in matrix A that span the null space. Null space of matrix A can be defined as the set of n column vectors x such that their […]

### Find an orthogonal basis for the column space of the matrix given below:

[ boldsymbol{ left[ begin{array}{ccc} 3 & -5 & 1 \ 1 & 1 & 1 \ -1 & 5 & -2 \ 3 & -7 & -8 end{array} right] }]This question aims to learn the Gram-Schmidt orthogonalization process. The solution given below follows the step-by-step procedure. In Gram-Schmidt orthogonalization, we assume the first basis vector […]

### Find the rate of change of f at p in the direction of the vector u.

[f(x,y,z) = y^2e^{xyz}, P(0,1,-1), u = <frac{3}{13},frac{4}{13},frac{12}{13}>] This question aims to find the rate of change or gradient and projections of vector spaces onto a given vector. Gradient of a vector can be found using following formula: [nabla f(x,y,z) = bigg ( frac{partial f}{partial x} (x,y,z),frac{partial f}{partial y} (x,y,z),frac{partial f}{partial z} (x,y,z) bigg )] Projection of a […]

### Let W be the set of all vectors of the form shown, where a, b, and c represent arbitrary real numbers

For the given set of all vectors shown as $W=left[ begin{matrix}4a + 3b\0\ begin{matrix}a+b+c\c – 2a\end{matrix}\end{matrix}right]$, and here a, b and c are arbitrary real numbers. Find vector set S which spans W or give an example to show that W is not a space vector. In this question, we have to find […]

### Find the values of x such that the angle between the vectors (2, 1, -1) and (1, x, 0) is 40.

The question aims to find the value of an unknown variable given in 3D vector coordinates and the angle between those vectors. The question depends on the dot product of two 3D vectors to calculate the angle between those vectors. As the angle is already given, we can use the equation to calculate the unknown […]