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Problem 192

# Find the dimension of the vector space spanned by: (i) $$(1,-2,3,-1)$$ and $$(1,1,-2,3)$$ (ii) $$(3,-6,3,-9)$$ and $$(-2,4,-2,6)$$ (iii) $$t^{3}+2 t^{2}+3 t+1$$ and $$2 t^{3}+4 t^{2}+6 t+2$$ (iv) $$t^{3}-2 t^{2}+5$$ and $$t^{2}+3 t-4$$ (v) $$\begin{array}{ll}11 & 2 \mid & \text { and } 11\end{array}$$ $$\begin{array}{ll}11 & 21\end{array}$$ (vi) $\begin{array}{rrrr} & 11 & 11 & \mid-3 & -3 \mid \\ & \mid-1 & -1 \mid & 3 & 3 \mid\end{array}$ (vii) 3 and $$-3$$.

Expert verified
The dimensions of the vector spaces spanned by the given sets are: (i) 2 (ii) 1 (iii) 1 (iv) 2 (v) 2 (vi) 2 (vii) 1
See the step by step solution

## Step 1: (i) Analyzing vectors

Analyzing the vectors: $$(1, -2, 3, -1)$$ and $$(1, 1, -2, 3)$$. We can see that neither of the vectors is a multiple of the other, so their span will have dimension 2.

## Step 2: (i) Conclusion

The dimension of the vector space spanned by $$(1, -2, 3, -1)$$ and $$(1, 1, -2, 3)$$ is 2.

## Step 3: (ii) Analyzing vectors

Analyzing the vectors: $$(3, -6, 3, -9)$$ and $$(-2, 4, -2, 6)$$. After close examination, we notice that the second vector is $$(-2/3)$$ times the first vector. This means that these vectors are linearly dependent and the span will only have dimension 1.

## Step 4: (ii) Conclusion

The dimension of the vector space spanned by $$(3, -6, 3, -9)$$ and $$(-2, 4, -2, 6)$$ is 1.

## Step 5: (iii) Analyzing polynomials

Analyzing the polynomials: $$t^3+2t^2+3t+1$$ and $$2t^3+4t^2+6t+2$$. We can see that the second polynomial is 2 times the first polynomial, so the polynomials are linearly dependent. Hence, the dimension of their span is 1.

## Step 6: (iii) Conclusion

The dimension of the vector space spanned by $$t^3+2t^2+3t+1$$ and $$2t^3+4t^2+6t+2$$ is 1.

## Step 7: (iv) Analyzing polynomials

Analyzing the polynomials: $$t^3-2t^2+5$$ and $$t^2+3t-4$$. In this case, neither polynomial can be expressed as a multiple of the other, so their span will have dimension 2.

## Step 8: (iv) Conclusion

The dimension of the vector space spanned by $$t^3-2t^2+5$$ and $$t^2+3t-4$$ is 2.

## Step 9: (v) Analyzing expressions

Analyzing the expressions: $$11 \text{ and } 2 \mid$$ and $$11 \text{ and } 21$$. Since the expressions are not linearly dependent, their span will have dimension 2.

## Step 10: (v) Conclusion

The dimension of the vector space spanned by $$11 \text{ and } 2 \mid$$ and $$11 \text{ and } 21$$ is 2.

## Step 11: (vi) Analyzing expressions

Analyzing the expressions: $11 \text{ and } 11 \mid -3 \text{ and } -3 \mid \text{ and } \mid{-1} \text{ and } {-1} \mid \\$ $$3 \text{ and } 3 \mid$$. These expressions are linearly dependent: the first and the third are equal, as are the second and the fourth. The set will have a dimension of 2.

## Step 12: (vi) Conclusion

The dimension of the vector space spanned by the given expressions is 2.

## Step 13: (vii) Analyzing numbers

Analyzing the numbers: 3 and $$-3$$. In this case, we have a one-dimensional vector space, so either 3 or $$-3$$ spans it. The span will have dimension 1.

## Step 14: (vii) Conclusion

The dimension of the vector space spanned by 3 and $$-3$$ is 1.

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