All Questions
Tagged with vector-spaces solution-verification
612
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Questions about how to show $d_1+\cdots +d_n-n+1 \leq {\text{dim}}_k k[x_1,\ldots,x_n]/\mathfrak{a}\leq d_1d_2\cdots d_n\quad $
The following are from Froberg's "Introduction to Grobner bases" , and Hungerford's undergraduate "Abstract Algebra" text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\...
- 3,683
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1
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71
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Showing that $k[x_1,\ldots,x_n]/\mathfrak{a}$ is a finite dimensional vector space over $k$ assuming basic linear algebra and min amount of abs alg.
The following are from Froberg's Introduction to Grobner bases, and Hungerford's undergraduate Abstract Algebra text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\backsimeq (k[x_1,\ldots,x_{n-1}])...
- 3,683
1
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1
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Proving the Set of Periodic Functions with Restrictions Form a Vector Space
I understand that a set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ cannot be a vector space because the set is not closed under the sum of the functions, as discussed here. However, I ...
- 115
3
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4
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Understanding the implication in linear algebra regarding vectors
Let $V$ be a subspace of $\mathbb{R}^n$ with the usual dot product, and let $\mathbf{z}, \mathbf{w} \in V$ be fixed vectors. If for every $\mathbf{v} \in V$ it holds that $\mathbf{z} \cdot \mathbf{v} =...
- 157
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1
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61
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Is there a finite dimensional vector space over a finite field with exactly two bases?
Is there a finite dimensional vector space over a finite field with exactly two bases?
I searched and found that the answer is NO. But I have an example that $\mathbb{Z}_{3}$ is a 1-dimensional vector ...
- 139
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1
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95
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How do I find maximal quotients of subspaces of the vector spaces $V_i$?
Say I have linear maps $V_1 \xrightarrow{h} V_2 \xleftarrow{g} V_3 \xrightarrow{f} V_4$. Assume $V_i$ are finite dimensional.
(1) I want to find maximal subspaces $W_i$ of $V_i$ such that in the ...
- 752
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63
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Critique my understanding of the determinant's relation to linear independence
If $det(A) = 0$ then the columns of $A$ are linearly dependent. This is a result that I could recite but struggled to reconcile after having completed my first course in elementary linear algebra.
...
- 197
1
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1
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Given an inconsistent overdeterminate system AX=b where $A\in M_{m×n}(R)$ and $b\in R^m$ with rank A=n. Find the least square approx. solution of AX=b
Suppose $A$ is a real matrix of order $m\times n$ with $m>n,b\in\Bbb R^m$ be such that the over determined system
of linear equations $AX=b$ is inconsistent and $\text{rank} (A)= n.$ Let $W$ be the ...
- 3,073
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1
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Consequences of definition of scalar product
Definition: Let $V$ be a vector space over the field $K=\mathbb{R}$ (or over $K=\mathbb{C})$. The scalar product on $V$ is a function $V\times V\to K,$ denoted by $(x,y)\mapsto \langle x,y\rangle$, ...
- 1,494
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1
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27
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Extending a Linearly Independent Set to form a Basis
Let $V= \mathbb R^4$. Consider the subspace $U=\{(a_1,a_2,a_3,a_4) \in \mathbb R^4 | a_1+a_2+a_3=0\}$ of $V$. We have $u_1=(0,0,0,1)$ and $u_2 = (5,-2,-3,0)$ of $U$. Find another $u_3 \in U$ such that ...
- 247
3
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104
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Proof for Vector Spaces $V = U + W$ with $U \cap W = {\mathbf{0}}$
I am going through question in my text book regarding proof. I have done that but I think I am making mistake of proving that $u_1=u_2$ and $w_1=w_2$ by using this same statement that I have to prove. ...
- 652
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(Proof verification) Prove that the set $\{w,v_1,v_2,...,v_r\}$ is linearly independent.
Problem: Let $S=\{v_1,v_2,...,v_r\}$ be a linearly independent subset of a vector space $V(F)$. If $w\in V$ such that $w\notin L(S)$, then prove that the set $S_1=\{w,v_1,v_2,...,v_r\}$ is also ...
0
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48
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Have I shown these linear functionals span the annihilator?
Full exercise.
Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$. Then
$$
\text{dim}\ U^0 = \text{dim}\ V - \text{dim}\ U.
$$
My Question.
My strategy is exactly how the first proof in ...
- 1,454
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45
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Let $S\subseteq V $ and $v\in V\setminus S$. Show that $S\cup \{v\}$ is l.d. $\iff v\in\langle S\rangle$
Let $V$ be a vector space, $S\subseteq V$ and $v\in V\setminus S$. Show that $S\cup \{v\}$ is linearly dependent if and only if $v\in\langle S\rangle$
So, can you help me evaluate my proof? Any ...
- 561
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189
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Stuck trying to prove $T = \alpha I$ for some $\alpha \in \mathbf{F}$.
Exercise.
Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Prove that $T$ has the same matrix with respect to every basis of $V$ if and only if $T$ is a scalar multiple of the identity ...
- 1,454