I am so unsure on where to even start at for these problems.
The first part is:
Given non-zero vectors $x$, $y$ in $R_n$, let $x\otimes y$ be the $n \times n$ matrix whose $(i,j)$ entry is $x_iy_j$. Find the dimension of the range of $x \otimes y$. Enter your answer when $n=21$.
Then (the second part) is asking: (Continuing from the first problem) In addition, $x, y$ are linearly independent. Let $A=x\otimes y + y\otimes x$ Find the dimension of the kernel of $A$. Enter your answer when $n=21$.
Defining $x$ as column $n\times 1$ matrix and $y$ as a row $1\times n$ matrix, the product $x\otimes y$ is simply the matrix product $xy$: $$ x\otimes y=xy=\begin{bmatrix}xy_1 &xy_2 & \cdots & xy_n \end{bmatrix}. $$ Notice that each column of this matrix is a multiple of $x$, then the dimension of the column space - the dimension of the matrix range, also known as the rank - is 1.
Another way to see this is recalling that the sum of the dimensions of the kernel and range for a matrix acting on a vector space is equal to the dimension of the vector space, what is called the rank-nullity theorem. The kernel of $xy$ is the space formed for all vectors orthogonal to $y$. If $y$ is nonzero and the vector space has dimension $n$, then the kernel has dimension $n-1$ and the range has dimension 1.
For the second part what you have is $A=xy+y^Tx^T$. As $x$ and $y$ are linearly independent, the kernel is the space formed by all vectors orthogonal to $x$ and $y$ simultaneously, a $(n-2)$-dimensional space.