Mart Hoert

2022-08-04

Let v be a vector over a field F with zero vector  0 and let s,T be a substance of V .then which of the following statements are false

Don Sumner

To evaluate the truth value of the statements, we need to consider the properties and definitions of vectors and subsets in a vector space. Let's examine each statement individually:
Statement 1: If $\mathbf{0}āS$, then $S$ is a subspace of $V$.
This statement is true. For $S$ to be a subspace of $V$, it must satisfy three conditions:
1. $\mathbf{0}āS$ (the zero vector is an element of $S$),
2. $S$ is closed under vector addition (if $\mathrm{š®}$ and $\mathrm{šÆ}$ are in $S$, then $\mathrm{š®}+\mathrm{šÆ}$ is also in $S$), and
3. $S$ is closed under scalar multiplication (if $\mathrm{š®}$ is in $S$ and $c$ is a scalar, then $c\mathrm{š®}$ is also in $S$).
Statement 2: If $S$ is a subspace of $V$, then $\mathbf{0}āS$.
This statement is true. As mentioned in the explanation of Statement 1, for $S$ to be a subspace of $V$, it must contain the zero vector $\mathbf{0}$.
Statement 3: If $SāŖT=\left\{\mathbf{0}\right\}$, then $S$ and $T$ are subspaces of $V$.
This statement is false. To demonstrate this, we can provide a counterexample. Consider $V$ as the vector space ${\mathrm{ā}}^{2}$ and let $S=\left\{\left(1,0\right)\right\}$ and $T=\left\{\left(0,1\right)\right\}$. Both $S$ and $T$ are subsets of $V$, and their union is $\left\{\left(1,0\right),\left(0,1\right)\right\}$, which does not equal $\left\{\mathbf{0}\right\}$. However, $S$ and $T$ are not subspaces of $V$ because they do not contain the zero vector $\mathbf{0}$.
To summarize:
- Statement 1 is true.
- Statement 2 is true.
- Statement 3 is false.

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