### Let f be a fixed 3×2 matrix, and H be the set of matrices A belonging to a 2×4 matrix. If we assume that the property FA = O holds true, show that H is a subspace of M2×4. Here O represents a zero matrix of order 3×4.

The aim of this question is to comprehend the key linear algebra concepts of vector spaces and vector subspaces. A vector space is defined as a set of all vectors that fulfill the associative and commutative properties for vector addition and scalar multiplication operations. The minimum no. of unique vectors required to describe a certain vector space is called basis vectors. A vector space is an n-dimensional space defined by linear combinations of basis vectors. Mathematically, a […]

### Find a basis for the eigenspace corresponding to each listed eigenvalue of A given below:

[ boldsymbol{ A = left[ begin{array}{cc} 1 & 0 \ -1 & 2 end{array} right], lambda = 2, 1 } ] The aim of this question is to find the basis vectors that form the eigenspace of given eigenvalues against a specific matrix. To find the basis vector, one only needs to solve the following system for $x$: [ A […]

### Find, correct to the nearest degree, the three angles of the triangle with the given vertices. A(1, 0, -1), B(3, -2, 0), C(1, 3, 3).

The main objective of this question is to find the three angles of a triangle given three vertices. The angles can be found using the dot product of the vectors representing the sides of the triangle. A triangle is a polygon with three-sides that is also referred to as a trigon. Every triangle possesses $3$ sides and […]

### Find the vectors T, N, and B at the given point. r(t)=< t^2,2/3 t^3,t > and point < 4,-16/3,-2 >.

This question aims to find the Tangent, Normal, and Binormal vectors by using the given point and a function. Consider a vector function, $vec{r}(t)$. If $vec{r}'(t)neq 0$ and $vec{r}'(t)$ exist then $vec{r}'(t)$ is called a tangent vector. The line that passes through the point $P$ and is parallel to the tangent vector, $vec{r}'(t)$, is the […]

### Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and area of the triangle PQR.

Take note of the following points: $P(1,0,1) , Q(-2,1,4) , R(7,2,7)$ Find a nonzero vector orthogonal to the plane through the points $P, Q$, and $R$. Find the area of the triangle $PQR$. The purpose of this question is to find an orthogonal vector and the area of a triangle using the vectors $P, Q,$ […]