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.
Explanations 
Textbooks 
Exams 
Magazine 
