Fact.
If $S=\{\mathbf{v}_1, \dots, \mathbf{v}_m\}$ is a spanning set of a subspace $V$ of $\R^n$, then any set of $m+1$ or more vectors of $V$ is linearly dependent.

Let $B’=\{\mathbf{w}_1, \mathbf{w}_2, \dots, \mathbf{w}_l\}$ be an arbitrary basis of the subspace $V$.
Our goal is to show that $l=k$.

As $B$ is a basis, it is a spanning set for $V$ consisting of $k$ vectors.
By the fact stated above, a set of $k+1$ or more vectors of $V$ must be linearly dependent.
Since $B’$ is a basis, it is linearly independent.
It follows that $l\leq k$.

We now change the roles of $B$ and $B’$.
As $B’$ is a basis, it is a spanning set for $V$ consisting of $l$ vectors.
So it follows from Fact that a set of $l+1$ or more vectors must be linearly dependent.
Since $B$ is a basis, it is linearly independent.
Hence $k \leq l$.

Therefore we have $l\leq k$ and $k \leq l$, and it yields that $l=k$, as required.

The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero
Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$.
Then prove that $V$ is a subspace of $\R^n$.
Proof.
To prove that $V=\{\mathbf{0}\}$ is a subspace of $\R^n$, we check the following subspace […]

If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent
Let $V$ be a subspace of $\R^n$.
Suppose that
\[S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}\]
is a spanning set for $V$.
Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.
We give two proofs. The essential ideas behind […]

Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)
Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.
Hint.
Recall that the rank of a matrix $A$ is the dimension of the range of $A$.
The range of $A$ is spanned by the column vectors of the matrix […]

Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis
Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient.
Let $W$ be the following subset of $P_3$.
\[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\]
Here $p'(x)$ is the first derivative of $p(x)$ and […]

Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix
Let \[A=\begin{bmatrix}
1 & 1 & 2 \\
2 &2 &4 \\
2 & 3 & 5
\end{bmatrix}.\]
(a) Find a matrix $B$ in reduced row echelon form such that $B$ is row equivalent to the matrix $A$.
(b) Find a basis for the null space of $A$.
(c) Find a basis for the range of $A$ that […]

[…] these into one matrix equation, we obtain [VA=U,] where Every Basis of a Subspace Has the Same Number of Vectors [A=begin{bmatrix} a_{1 1} & a_{1 2} & cdots & a_{1 k} \ a_{2 1} & a_{2 2} […]

## 1 Response

[…] these into one matrix equation, we obtain [VA=U,] where Every Basis of a Subspace Has the Same Number of Vectors [A=begin{bmatrix} a_{1 1} & a_{1 2} & cdots & a_{1 k} \ a_{2 1} & a_{2 2} […]