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The first four Hermite polynomials are $1,2 t,-2+4 t^{2},$ and $-12 t+8 t^{3} .$ These polynomials arise naturally in the study of certain important differential equations in mathematical physics. $^{2}$ Show that the first four Hermite polynomials form a basis of $\mathbb{P}_{3}$ .

Since the vectors form a basis for $\mathbb{R}^{4},$ the polynomials must $\operatorname{span} \mathbb{P}_{3}$ .

Calculus 3

Chapter 4

Vector Spaces

Section 5

The Dimension of a Vector Space

Vectors

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work, even Pelino meal overseas. One, then one minus t then to minus 40 plus T square. The last one is six minus 18 T. That's nine. T squared us. Sorry. Minus cube. Okay, so in terms of the Matrix we have of four by four matrix. So it is the first efforts column represented a constant term. Off off. Pretty nami second column represent our one and so on and so forth. We have off one are off to and our off three. So for the person in question, we only have one and other things will be zero. The second equation. We have one constant term and, um, negative one for that t and also right. There are two zeros and the third column. We have two for constant term and negative war or tea, one for t squared. And for the last column, we have six accounts in term negative 18 for teeth and nine for t squared. And next one or T. Cute. Okay, so it's no hard to observe that they are four people. Two columns. So this four is exactly, um excuse me. So these four people columns, uh, exactly equal to the number off. Go to the number. Oh, pretty no meals given complete onions given in our meals. Nam Joo's. So that means these columns for columns are columns. All right, the nearly independent. Okay, so the answer is yes. This is four. Really? Nah. Mills will spend the whole p three. So? So these four? Well, you know, meals lean on mules. It's been the three, so we're

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