[Math] Linear Algebra Vector True and False Questions

linear algebravector-spaces

I have a few true and false questions. I have explanations for them could someone please check them over?

  1. $R^3$ contains two disjoint subspaces. I think this is true for example {1,2,3} and {4,5,6} are r3 but have nothing in common.

  2. any 4 dimensional subspace of $L^{20}$ is isomorphic to R4. I think this is true because I can't find a set of vectors that doesn't satisfy f(a+b)= f(a)+f(b) and f(ab)=af(b)

3.A is a vector space and H=span{v1,v2,v3…vX} the dimension of A has to be X. I think this is false but I can't explain it.

4.Every linear transformation from r2 to r3 can be represent as any sort of transofrmation matrix. I think this is true because anything can be written as a matrix.

5.If Q is a 5 dimensional vector space. For every W greater than or equal to 5, there there is some k dimensional vector space E so that Q is isomorphic to subspace of E. I think this is true because k has to be greater than 5 so 5 will fall under a k dimension.

Best Answer

  1. False. Note that every subspace contains the point 0.

  2. True. Use the more general fact that every vector space (over $\mathbb{R}$) of dimension $n$ is isomorphic to $\mathbb{R}^n$. If you haven't come across this theorem yet, try to prove it in your special case of $n=4$. (Take an arbitrary 4-dimensional vector space, and consider a basis for that space. Map the basis vectors onto the standard basis for $\mathbb{R}^4$. Why is this an isomorphism?).

  3. False. Hint: What is the dimension of $\text{span}\{(0,1), (0,2)\}$? (The answer is not 2. Can you generalise by finding a condition on the $x_i$ that makes statement 3 true? Think of the definition of a basis.)

  4. True. It's true in general that every linear transformation between vector spaces can be expressed as a matrix.

    Take a basis $B_1$ of $\mathbb{R}^2$, and a basis $B_2$ of $\mathbb{R}^3$. Call your linear transformation $f$. The coefficients of the matrix you want (call it $F$) will be the image of the basis vectors in $B_1$, in terms of the basis vectors in $B_2$. Try to show as explicitly as possible that, $Fx=f(x)$ for every vector $x$ in $\mathbb{R}^2$ (the domain).

    In my opinion, the process of finding this matrix is quite confusing until you've seen it done many times (I have only provided a sketch of how it is done above). Perhaps you may wish to come back to this when you have a bit more experience under your belt.

  5. True. This is similar to statement 2. You will need the fact that, for every $n \in \mathbb{N}$, there exists a vector space of dimension $n$. Can you see an obvious example? You will also need to be able to find a 5-dimensional subspace of a higher-dimensional vector space. Hint: take a basis, and consider a subset of that basis. How can you turn it into a subspace, and how can you control the dimension of your subspace?

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