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My question is Determine whether the polynomial $p(t) = t^3-3t^2+5t+1, q(t) = t^3-t^2+8t+2, r(t) = 2t^3-4t^2+9t+5$ in the vector space of all polynomials of degree less than or equal to 3 are linear independent.

Following show my try:enter image description here

How to show " all polynomials of degree less than or equal to 3 are linear independent" this part

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    $\begingroup$ Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers. – For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. $\endgroup$
    – Martin R
    Dec 6 '21 at 4:12
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    $\begingroup$ You had four equations. How did that become three? Otherwise, your approach from definition is correct, and the equations are correct. If you removed the fourth equation because you guessed that it did not matter, then you're correct, of course. $\endgroup$ Dec 6 '21 at 4:49
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First of all, your work is correct and you showed that the $3$ polynomials $p,q,r$ are linearly independent. But the second part that you wanted to show that "all polynomials of degree less than or equal to 3 are linear independent" cannot be true. Look at the example $p = t^3, q = t^2, r = t^3+t^2 \implies -r + p + q = 0\implies p,q, r$ are linearly dependent. But all polynomials of degree less than or equal to $3$ form a vector space.

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