Linear Algebra Subspace of Polynomial Question Help please Watch

NoahMal
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Any hints on this question would be greatly appreciated
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NotNotBatman
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You can write the equations in matrix form and show that it is linearly independent, by calculating to determinant of the matrix with constant coefficients.
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NoahMal
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(Original post by NotNotBatman)
Looks like the other thread was deleted without my reply being restored.

You can write the equations in matrix form and show that it is linearly independent, by calculating to determinant of the matrix with constant coefficients.
How do I write the equations in matrix form?
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NotNotBatman
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(Original post by NoahMal)
How do I write the equations in matrix form?
 \begin{pmatrix} 1 & -1 & 0\\  1 &0  &-1 \\   0&1  & 0 \end{pmatrix}  \begin{pmatrix} 1\\  t\\  t^2 \end{pmatrix}

Start with writing the degrees of t as a column vector and then in the 3x3 matrix, write the coefficients of the equations along the rows.
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NoahMal
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(Original post by NotNotBatman)
 \begin{pmatrix} 1 & -1 & 0\\  1 &0  &-1 \\  0&1  & 0 \end{pmatrix}  \begin{pmatrix} 1\\  t\\  t^2 \end{pmatrix}

Start with writing the degrees of t as a column vector and then in the 3x3 matrix, write the coefficients of the equations along the rows.
Ok I found the Determinant =1 therefore linearly independant and whats the next step ? Finding this really quite confusing
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NoahMal
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(Original post by NotNotBatman)
 \begin{pmatrix} 1 & -1 & 0\\  1 &0  &-1 \\  0&1  & 0 \end{pmatrix}  \begin{pmatrix} 1\\  t\\  t^2 \end{pmatrix}

Start with writing the degrees of t as a column vector and then in the 3x3 matrix, write the coefficients of the equations along the rows.
Here is what I've done so far
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NoahMal
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Here are the co-ordinates I have found
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NoahMal
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DFranklin
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(Original post by NoahMal)
Any hints on this question would be greatly appreciated
(Original post by NotNotBatman)
Looks like the other thread was deleted without my reply being restored.

You can write the equations in matrix form and show that it is linearly independent, by calculating to determinant of the matrix with constant coefficients.
So: I don't generally like the "oh, do the matrix thing and calculate the determinant"; it depends on how your course is structured, but often you haven't covered determinants rigourously (i.e. in a pure maths course) at the point you're answering questions like this. And it's totally a "black box": you don't get any extra information about how the system behaves, which leaves you at square one for any subsequent questions.

In this case, given the second part, it really doesn't seem the best plan. It is very easy to write 1, t and t^2 in terms of p1, p2 and p3. This immediately shows that they span (and therefore must be a basis since the number of vectors = the dimension of the space). And it also makes writing q1 and q2 in terms of p1, p2 and p3 straightforward.
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