Assignments 2024 Summer Bb 190627904 Math 156 Homework 3 - Soluti 190627904 Nathanson math lectures - You X A Student Dashboard + learn-us-east-1-prod-fleet02-xythos.content.blackboardcdn.com/61aab133e7df2/190627904?X-Blackboard-S3-Bucket-learn-us-east-1-prod-fleet01-xythos&X-Blackboard-Expiration=17174376000... 10-0 14 レーローヒー 900-0 00-1 0600 100-1 <-4- 17 / 41 100% H 10 +420 TTANO and so L is a vector subspace if and only if 0 L if and only if b = 0. This completes the proof. Exercises. (1) Compute the following linear combinations of vectors: (a) In the vector space R², 76)-5(9 (b) In the vector space R³, ☐ (2) +25 6 +38 9 3 (c) In the vector space R4, 15 16 100 0-0-0 2 5 3 -6 0 5 2 3 11 6 (2) Compute the following linear combinations of vectors: (a) In the vector space R², 3 ³ (2)-8(1) (b) In the vector space R³, 3 ·0-0-0 3 1 -5 3 (c) In the vector space R5, 2 +7 1 4 3 7 -8 ·0-0-0 8 1 +3 2 3 (3) In the vector space R², let 1 -2 1 9 17 and e₁ = and e2 = Guest 。 L ... and £ (1) Jun 3 12:34 Assignments 2024 Summer Bb 190627904 190627904 Math 156 Homework 3 - Soluti Nathanson math lectures - You X A Student Dashboard + learn-us-east-1-prod-fleet02-xythos.content.blackboardcdn.com/61aab133e7df2/190627904?X-Blackboard-S3-Bucket-learn-us-east-1-prod-fleet01-xythos&X-Blackboard-Expiration=17174376000... 5 / 41 100% + | A This completes the proof. Exercises. (1) Compute the solution space in R² of the following linear equations. Write the solutions as vectors in R². 3 5 6 (a) (b) (c) (d) x+y=0. x+y=3. x+3y=1 2x-7y=-3 (2) Compute the solution space in R3 of the following linear equations. Write the solutions as vectors in R³. 6 (a) (b) (c) (d) O MELVYN B. NATHANSON x+y+z 0. x+y+z -7. x+3y 7z 1 5x 2y+3x=-11. 。 L Guest ... Jun 3 12:22 (8) Graph the solution spaces of the following equations in R2: (a) LINEAR EQUATIONS - 10x 9y=15 (b) (c) 10x = = 15 -9y= 15 18 (3) In the vector space R², let e₁ = and e2 = and f₁ = (+) and f₂ = MELVYN B. NATHANSON 2 (a) Write the vectors f₁ and f₂ as linear combinations of the vectors e₁ and e2. (b) Write the vectors е₁ and e2 as linear combinations of the vectors f₁ and f2. (c) Prove that every vector x = (21) € R² E R2 has a unique representation as a linear combination of the vectors f₁ and fo. (4) (a) In the vector space R³, let 1 0₁ = 0 02 = e3 = 0 (b) 1 ER³ Prove that every vector x = 22 13 has a unique representation as a linear combination of the vectors e1, e2, and e3. In the vector space R3, let -(0) f₁ = 1 1 1 = 1 -(0)-(0) f₂ = Prove that every vector x = 11 12 13 f3 R3 has a unique representation as a linear combination of the vectors f₁, f₂, and f3.