Problem

Determine whether the given set $S$ of vectors is…

02:06

Problem 2 Easy Difficulty

Determine whether the given set $S$ of vectors is closed under addition and closed under scalar multiplication. In each case, take the set of scalars to be the set of all real numbers.
The set $S:=U_{n}(\mathbb{R})$ of all upper triangular $n \times n$ matrices with real elements.

Answer

scalar multiplication.

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Calculus 3

Differential Equations and Linear Algebra

Chapter 4

Vector Spaces

Section 2

Definition of a Vector Space

Discussion

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Video Transcript

All right, so this time our vector set is the set trying to right here the set of a lot upper triangular matrix sees with riel elements in their end by end. So I'm just gonna pick an end like four. And I'm gonna put in a B c d zero e f g zero age no hay 00 h i 000 j. So there we have and upper Triangular matrix. So is it closed under addition? So let's say that I took another matrix. Okay? Come on. L m n 00 p Q 00 Ah, zero. Wait. I got something wrong, you know? Oh, my oh, is messing me up? Hey, um, are s 000 t noticing that this is an O. Okay, So if we add those together, um, we are going to get all real numbers. Um, in the upper triangle of the matrix and all zeros in the lower, they're gonna look like a plus K B plus L C plus m and then zero plus zero down here in the bottom Left is gonna be zero. So yes, it is closed. Um, So is it closed under scaler? multiplication. So I'm going to erase here and we'll think about scaler. Multiplication. So if I multiply this by any real number, let's say okay, Okay, well, that's going to give me que times a k a times b k time. See? And these are all real numbers. And then, Kate, time zero is just gonna be zero. So again, Yes, it is closed.

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