[Math] Finding which sets are subspaces of R3

invariant-subspacelinear algebravector-spacesvectors

Image of the question

Hello. I have attached an image of the question I am having trouble with. I thought that it was 1,2 and 6 that were subspaces of $\mathbb R^3$.

Here is my working:

  1. Rearranged equation —> $x+y-z=0$. Is a subspace since it is the set of solutions to a homogeneous linear equation.

  2. $0$ is in the set if $x=y=0$. Is a subspace. (I know that to be a subspace, it must be closed under scalar multiplication and vector addition, but there was no equation linking the variables, so I just jumped into thinking it would be a subspace…)

  3. Rearranged equation —> $xy – xz=0$. $0$ is in the set if $x=0$ and $y=z$. I said that $(1,2,3)$ element of $R^3$ since $x,y,z$ are all real numbers, but when putting this into the rearranged equation, there was a contradiction. So, not a subspace.

  4. = space $\{\,(1,0,0),(0,0,1)\,\}$. set is not a subspace (no zero vector)

  5. Similar to above.

  6. $0$ is in the set if $m=0$. Again, I was not sure how to check if it is closed under vector addition and multiplication. From seeing that $0$ is in the set, I claimed it was a subspace.

I appreciate any help. Thank you.

Best Answer

Here are the definitions I think you are missing:

Closure under vector addition:

A subset $S$ of $\mathbb{R}^3$ is closed under vector addition if the sum of any two vectors in $S$ is also in $S$. In other words, if $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are in the subspace, then so is $(x_1+x_2,y_1+y_2,z_1+z_2)$. For example, if we were to check this definition against problem 2, we would be asking whether it is true that, for any $x_1,y_1,x_2,y_2\in\mathbb{R}$, the vector $(x_1,y_2,x_1y_1)+(x_2,y_2,x_2y_2)=(x_1+x_2,y_1+y_2,x_1x_2+y_1y_2)$ is in the subset.

Closure under scalar multiplication:

A subset $S$ of $\mathbb{R}^3$ is closed under scalar multiplication if any real multiple of any vector in $S$ is also in $S$. In other words, if $r$ is any real number and $(x_1,y_1,z_1)$ is in the subspace, then so is $(rx_1,ry_1,rz_1)$. For example, if we were to check this definition against problem 2, we would be asking whether it is true that, for any $r,x_1,y_1\in\mathbb{R}$, the vector $(rx_1,ry_2,rx_1y_1)$ is in the subset.

The set $\{s(1,0,0)+t(0,0,1)|s,t\in\mathbb{R}\}$ from problem 4 is the set of vectors that can be expressed in the form $s(1,0,0)+t(0,0,1)$ for some pair of real numbers $s,t\in\mathbb{R}$. A similar definition holds for problem 5.

(Also I don't follow your reasoning at all for 3.)

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