Let V be a vector space over a field of characteristic not equal totwo. (a) Let
ID: 2937210 • Letter: L
Question
Let V be a vector space over a field of characteristic not equal totwo. (a) Let u and v be distinct vectors in V. Prove that{u,v} is linearly independent if and only if {u+v,u-v} is linearlyindependent. (b) Let u,v, and w be distinct vectors in V. Prove that{u,v,w} is linearly independent if and only if {u+v,u+w,v+w} islinearly independent.Please explain with steps. Will rate Lifesaver. Thankyou (a) Let u and v be distinct vectors in V. Prove that{u,v} is linearly independent if and only if {u+v,u-v} is linearlyindependent. (b) Let u,v, and w be distinct vectors in V. Prove that{u,v,w} is linearly independent if and only if {u+v,u+w,v+w} islinearly independent.
Please explain with steps. Will rate Lifesaver. Thankyou
Explanation / Answer
QuestionDetails: (a) Let u and v be distinct vectors in V. Prove that{u,v} is linearly independent if and only if{u+v,u-v} is linearly independent.
PROPOSITION
U,V ARE L.I.
TPT....U+V,U-V ARE L.I.
LET
P[U+V]+Q[U-V]=0
U[P+Q]+V[P-Q]=0
SINCE U AND V ARE INDEPENDENT
P+Q=0
P-Q=0
ADDING
2P=0....P=0....SO....Q=0
THAT IS P[U+V]+Q[U-V]=0...IMPLIES P=Q=0...SO...(U+V) AND (U-V) ARL.I
CONVERSE
(U+V) AND (U-V) ARE L.I
TPT U AND V ARE INDEPENDENT
P[U+V]+Q[U-V]=0...IMPLIES P=Q=0
U[P+Q]+V[P-Q]=0
UR+VS=0....WHERE R=P+Q...AND ...S=P-Q
BUT P=Q=O...SO R=0 AND S=0
THAT IS UR+VS=0 IMPLIES R=S=0...SO U AND V ARE L.I....PROVED
(b) Let u,v, and w be distinct vectors in V. Prove that{u,v,w} is linearly
independent if and only if {u+v,u+w,v+w} is linearlyindependent. YOU CAN PROCEED IN THE SAME WAY AS ABOVE TAKING
P[U+V]+Q[U+W]+R[V+W]=0...ETC..TRY...
IF IN DIFFICULTY PLEASE COME BACK
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