MATH136 Lecture 2: Week 2 (lec 4-6) Lecture Notes

• School: University of Waterloo
• Department: Mathematics
• Course: MATH136
• Professor: Patrick Roh

Document Summary

If for any then is closed under addition. (note: this is property 1 of ). If for any and then is closed under scalar multiplication. (property 6) If a non-empty subset of satisfies the ten properties of then it is a subspace of . If and for all and , then is a subspace of . Properties 2, 3, 7, 8, 9, 10 must hold because we are applying the same operations as in . Properties 1 and 6 are what is being checked. Note: our proof shows that any set that does not contain the zero vector is not a subspace. is not a subspace because . is a plane through the point. By the subspace test, is a subspace of . Note: a line through the origin with direction vector. Given vectors the dot product of and is. The dot product is also referred to as the standard inner product or the scalar product (when in ).