hi all,
I can't seem to crack the code for this series of true and false questions, I was wondering if someone with adequate knowledge pertaining to linear/matrix algebra could help me, thanks!
1. The weights c1,…,cp in a linear combination c1v1+⋯+cpvp cannot all be zero.
2. The solution set of the linear system whose augmented matrix is [a1a2a3|b] is the same as the solution set of equation x1a1+x2a2+x3a3=b.
3. If u and v are non-parallel, then the set Span{u,v} is always visualized as a plane through the origin.
4. When u and v are non-parallel vectors, Span{u,v} contains only the line through u and the origin, and the line through v at the origin.
5. If a is in Span{b,c}, then b is in Span{a,c}
Much appreciated!
I can't seem to crack the code for this series of true and false questions, I was wondering if someone with adequate knowledge pertaining to linear/matrix algebra could help me, thanks!
1. The weights c1,…,cp in a linear combination c1v1+⋯+cpvp cannot all be zero.
2. The solution set of the linear system whose augmented matrix is [a1a2a3|b] is the same as the solution set of equation x1a1+x2a2+x3a3=b.
3. If u and v are non-parallel, then the set Span{u,v} is always visualized as a plane through the origin.
4. When u and v are non-parallel vectors, Span{u,v} contains only the line through u and the origin, and the line through v at the origin.
5. If a is in Span{b,c}, then b is in Span{a,c}
Much appreciated!
-
1. False. They could all be 0 if we wanted, giving us a trivial way to obtain a linear combination of the zero vector.
2. True. This is converting between different notations.
3. True, since u and v are linearly independent. This is assuming that u and v are typical vectors that belong to the same vector space R^n, as opposed to some other type of vector space, like polynomials.
4. False. Span{u, v} also contains vectors on the entire plane that spans both lines, containing vectors such as 2u + 3v.
5. False. Suppose a = (0, 2) and b = (1, 0) and c = (0, 1).
Then a is in Span{b, c}, since a = 0b + 2c.
But b is NOT in Span{a, c} = Span{c}.
2. True. This is converting between different notations.
3. True, since u and v are linearly independent. This is assuming that u and v are typical vectors that belong to the same vector space R^n, as opposed to some other type of vector space, like polynomials.
4. False. Span{u, v} also contains vectors on the entire plane that spans both lines, containing vectors such as 2u + 3v.
5. False. Suppose a = (0, 2) and b = (1, 0) and c = (0, 1).
Then a is in Span{b, c}, since a = 0b + 2c.
But b is NOT in Span{a, c} = Span{c}.